Method and Apparatus for Generation of Luck and Skill Scores

ABSTRACT

The present invention generally relates to games that involve luck and skill, such as poker. Specifically, the subject invention provide means, method and apparatus for the generation of statistics relating to a player&#39;s luck and skill as exhibited in prior games (“luck and skill statistics” or “luck and skill scores”). In the preferred embodiments, statistics or scores are generated for participants in a poker game. These statistics quantify how lucky or skillful a player has been over a given period of time. The data can be used to enhance the experience of the viewing public, and to aid a player&#39;s self-assessment.

The present invention generally relates to games that involve luck andskill, such as poker. Specifically, the subject invention provide means,method and apparatus for the generation of statistics relating to aplayer's luck and skill as exhibited in prior games (“luck and skillstatistics” or “luck and skill scores”). In the preferred embodiments,statistics or scores are generated for participants in a poker game.These statistics quantify how lucky or skillful a player has been over agiven period of time. The data can be used to enhance the experience ofthe viewing public, and to aid a player's self-assessment.

BACKGROUND

Interest in playing and viewing poker has exploded in the last severalyears. Watching poker on television is more enjoyable now than in thepast because there are now miniature cameras installed at the card tablewhich allow the home viewer to see a player's hole cards, which arehidden from the view of his opponents. The player's hole cards aretypically displayed on the screen, along with the percentage chance thathe will win the hand. As subsequent cards are dealt, these percentagesare updated. What makes this exciting is that the announcer can thenobserve, “John bluffed Greg and got Greg to fold a hand that was a threeto one favorite to win, what an aggressive move!” or “John took a really‘bad beat’ in that hand because Greg's ‘miracle card’ got dealt,allowing Greg to win the hand even though John was a 20:1 favorite towin.” In short, exposing the game to viewer scrutiny makes it moreinteresting.

Poker luck and skill statistics would similarly enhance the experienceof the poker game by providing additional statistical informationregarding the strength of a player's cards and a player's strategy. Itis analogous to the idea of having baseball statistics, likeruns-batted-in and on-base percentage. For example, whenever a playerwins a World Series of Poker event, the question always arises: Did heget lucky, or was it skill? The question is particularly pertinent whenthe winner is an amateur and not a professional poker player.Conventional wisdom states that in order to win a particular pokertournament, even a skillful player must also get lucky. While thisassessment, which is shared by professional players, is likely to bequalitatively correct, it is not particularly illuminating because itdoes not answer the questions, “How lucky does the player need to be?”and “How lucky was the winner of that particular game?” To answer thesequestions, the concept of poker luck and skill scores are used, formulaswhich provide a quantitative index as to how lucky or skillful a playerhas been over a given period of time. These formulas are specificallydetailed in the next section.

Poker luck and skill statistics can be used by both the poker player andthe poker game viewer. The poker viewer's enjoyment of the game isenhanced in the same way that a baseball viewer's enjoyment of the gameis enhanced by baseball statistics. For example, when the players' chipstacks are displayed on the screen, their poker luck and skillstatistics can be displayed as well. The announcer could then comment,“Greg is up $100,000 in the cash game so far. His cards have not beenlucky but he's made up for it by playing well.” Or, “Phil is knocked outthe tournament complaining about his unlucky bad beats, but we see fromhis luck score that his cards have actually been quite lucky.” A pokerfan might attach more significance to wins achieved in the face oflackluster luck scores and high skill scores, and the merits of winsachieved with unusually good luck scores and poor skill scores could bedebated.

Additionally, the poker player can use his own poker luck and skillstatistics to improve his play. For example, an internet player mightobserve that using a certain style of play, his success is dependent, orindependent, of the quality of his luck score. After a successful game,a player could determine how much his skill reading and bluffingopponents contributed to his victory, and how much his lucky cardsplayed a role.

For a better understanding of the present invention, reference is madeto the following description taken in conjunction with the examples.

DESCRIPTION OF THE INVENTION AND EMBODIMENTS

Preferred embodiments of the invention are presented below for thepurposes of illustration and description. These embodiments arepresented to aid in an understanding of the invention and are notintended to, and should not be construed to, limit the invention in anyway. All alternatives, modifications and equivalents that may becomeobvious to those of ordinary skill upon a reading of the presentdisclosure are included within the spirit and scope of the presentinvention. Additionally, the present disclosure is not a primer on gamesof luck and skill, nor on computer software, systems or apparatus forimplementing the methods described herein. Basic concepts known to thoseskilled in the industry have not been set forth in detail.

The Primary Luck and Skill Scores—Preferred Area

The primary luck and skill scores are elegant and simple enough for youraverage poker enthusiast to understand intuitively. They make use ofstatistical data which is already commonly displayed on poker programs.The following formulas apply to a game of Texas Hold 'Em, although theyare easily generalized to any poker variant.

Let

-   V₀=expected value before blinds and antes are posted and before hole    cards dealt-   V₁=expected value before action preflop (after blinds and antes    posted and after hole cards dealt)-   V₂=expected value after action preflop-   V₃=expected value before action postflop-   V₄=expected value after action postflop-   V₅=expected value before action on the turn-   V₆=expected value after action on the turn-   V₇=expected value before action on the river-   V₈=expected value after action on the river (the amount actually won    or lost in the hand)-   C=probability of the player winning the hand-   P=pot size-   B=amount put in the pot by the player-   L=luck score for the hand-   S=skill score for the hand-   N=normalization factor

Note that N can be set equal to 1, P₁ (the total value of the blinds andantes), the total amount of the stacks of all the players at the table,or even to the total amount of the stacks of all the players (in atournament structure.)

Note that V₀=P₀=0.

Then

$\begin{matrix}{V_{i} = {{C_{i}P_{i}} - B_{i}}} & (1) \\{L = \frac{( {V_{1} - V_{0}} ) + ( {V_{3} - V_{2}} ) + ( {V_{5} - V_{4}} ) + ( {V_{7} - V_{6}} )}{N}} & (2) \\{S = \frac{( {V_{2} - V_{1}} ) + ( {V_{4} - V_{3}} ) + ( {V_{6} - V_{5}} ) + ( {V_{8} - V_{7}} )}{N}} & (3) \\{{L + S} = {\frac{V_{8} - V_{0}}{N} = \frac{V_{8}}{N}}} & (4)\end{matrix}$

Let us examine a specific example, using a normalization factor N=P₁.Suppose that the blinds are $50-$100 and the table is three-handed withPlayer X in the small blind, Player Y in the big blind, and Player Z onthe button. Before the hole cards are dealt, each player has an equalprobability of 1/3 of winning the hand. After the blinds are posted andafter the hole cards are dealt, but before betting, the probabilities ofX, Y, and Z winning the hand are 40%, 20%, and 40%, respectively. Zcalls $100, X folds, and Y checks. With X folding, the probabilities ofX, Y, and Z winning the hand change to 0%, 40%, and 60%, respectively.After the flop is dealt, the probabilities of Y and Z winning the handchange to 20% and 80%, respectively. Y bets $200, Z raises to $400, andY calls $200. After the turn card is dealt, the probabilities of winningfor Y and Z change to 10% and 90%. Y and Z check the turn. After theriver card is dealt, the probabilities for Y and Z winning change to100% and 0% when Y completes his draw. Y bets $1000, and Z calls. Usingequations (1)-(4), we can then generate the following analysis for thishand:

Player X Player Y Player Z Player X Player Y Player Z Player X Player YPlayer Z i C_(i) C_(i) C_(i) B_(i) B_(i) B_(i) P_(i) V_(i) V_(i) V_(i) 00.33 0.33 0.33 0 0 0 0 0 0 0 1 0.4 0.2 0.4 50 100 0 150 10 −70 60 2 00.4 0.6 50 100 100 250 −50 0 50 3 0 0.2 0.8 50 100 100 250 −50 −50 100 40 0.2 0.8 50 500 500 1050 −50 −290 340 5 0 0.1 0.9 50 500 500 1050 −50−395 445 6 0 0.1 0.9 50 500 500 1050 −50 −395 445 7 0 1 0 50 500 5001050 −50 550 −500 8 0 1 0 50 1500 1500 3050 −50 1550 −1500 L S Player X0.04 −0.40 Player Y 4.80 5.53 Player Z −4.87 −5.13

Let us examine the results of this analysis to gain an intuitiveunderstanding of the mathematical formulation of the luck score L andthe skill score S. Note that for purposes of analysis, after a playerfolds his hand (as player X did), C_(i) is set to zero because theplayer can no longer win the hand, and B_(i) holds a constant valuebecause the player can no longer change the amount he has already putinto the pot. We then observe that the net amount actually lost or wonby a player in a given hand is given by the value V₈, the expected valueafter the action on the river is complete. The question we seek toanswer, then, is what portion of V₈ was obtained by luck, and whatportion was obtained by skill? As a hand progresses, like in the exampleabove, the expected value for a given player will change as a result oftwo processes: cards are dealt, and action (checking, betting orfolding) is taken. In short, the changes in expected value that arise asa result of cards being dealt are attributed to luck, while changes inexpected value that arise as a result of players' unforced action areattributed to skill. This statement is captured by equations (2) and(3). Note that in equations (2) and (3), the changes in expected valueare divided by N. In the example above, N is set equal to P₁, the totalvalue of the blinds and antes. This is done to “normalize” thestatistics, so that a meaningful comparison of luck and skill scores canbe made between hands in different games, or between hands which occurearly and late in a tournament. In other words, the amount won or loston a given hand is considered relative to the size of the blinds andantes. Alternatively, by changing the value of N, the amount won or loston the hand can be considered relative to the total amount of the stacksof all the players at the table, or to the total amount of the stacks ofall the players in a tournament structure. N can also be set to 1 sothat the statistics are not normalized.

Equation (4) demonstrates that V₈/N, the normalized amount won (or lost)by the player in the hand, is the sum of the normalized amountattributable to luck and the normalized amount attributable to skill. Inthe example above, player Y's $1,550 win is attributable to both goodluck (L=4.80) and skill (S=5.53), with slightly more skill than luck.Player Z's $1,500 loss is attributable to both bad luck (L=−4.87) andlack of skill (S=−5.13), with slightly more lack of skill than bad luck.Player X's $50 loss is attributable to good luck (L=0.07) with a greaterlack of skill (S=−0.40). Understandably, the amounts of luck and skillinvolved in Player X's small loss are orders of magnitude less than theamounts involved with Player Y's larger win and Player Z's larger loss.

Now that we have examined the luck score and skill score for a givenhand, what is the overall luck score l and overall skill score s for agiven number of hands over a given time period? There are two ways thiscan be reported. The overall scores for a given number of hands can beeither: the average of the scores for each individual hand,

l= L  (5)

s= S  (6)

or the sum of the scores for each individual hand,

$\begin{matrix}{l = {\sum\limits_{\underset{hands}{all}}\; L}} & (7) \\{s = {\sum\limits_{\underset{hands}{all}}\; S}} & (8)\end{matrix}$

A few comments about the luck and skill scores are in order. It isimportant to note that the way luck and skill are calculated does notimply that the best strategy is to maximize expected value for each andevery given hand. Rather, the best strategy is to maximize expectedvalue over the game, which encompasses all the hands. So, for example,an aggressive player might incur a negative skill score for a given handwhile deliberately creating a table image. However, this move mightallow him to maximize his skill score for a later, larger pot when hisopponents don't give him credit for a premium hand.

There are many intuitive and practical advantages to calculating theluck and skill scores in the way described above. First, the luck andskill scores are calculated using information which is already displayedto the poker television viewer: percentage chance of winning, pot size,and amount each player is putting in the pot. Second, the scores givemathematical validity to the intuitive concept that a skilled pokerplayer will “get his money in with the best of it”; in other words,increase the pot size when he is statistically favored to win. When aplayer holding the worse hand bluffs another player out of a pot, thisis reflected positively in the bluffer's skill score and negatively inthe loser's skill score. When a player “sucks out” on the river, this isreflected positively in his luck score and negatively in his opponent'sluck score. A final advantage to calculating luck and skill scores inthis way is that knowledge of the board cards that would have been dealthad players stayed in the hand until showdown is not required. If aplayer folds before the flop, for example, whether or not he would havehad the best hand on the river does not affect his luck or skill scores.

It should be noted that the formula for the expected value V is easilyadjusted to the situation in which there is a chance of a split pot.Equation (1) then becomes

$\begin{matrix}{V = {( {{CP} - B} ) + \frac{e\; P}{f}}} & (9)\end{matrix}$

Where

-   e=the probability of the player winning a split pot-   f=the number of players sharing in the split pot-   C=probability of the player winning the pot (without splitting)-   P=pot size-   B=amount put in the pot by the player

The expected value V is also easily adjusted to the situation in whichthere is a side pot. Equation (1) then becomes

V=(CP−B)+(cp−b)  (10)

Where

-   C=the probability of winning the main pot-   P=size of the main pot-   B=amount put into the main pot by the player-   c=the probability of winning the side pot-   p=the size of the side pot-   b=amount put into the side pot by the player

The luck and skill scores as described above are also easily generalizedto any poker variant with different numbers of streets or cards.Equations (2) and (3) then become

$\begin{matrix}{L = {\sum\limits_{\underset{streets}{all}}\; T}} & (11) \\{S = {\sum\limits_{\underset{streets}{all}}\; U}} & (12)\end{matrix}$

Where

-   T=change in expected value V on a given street as a result of a card    (or cards) being dealt, or as a result of forced bets (blinds and    antes)-   U=change in expected value V on a given street as a result of    players' unforced action, after all action is complete on that    street

The Secondary Luck and Skill Scores—Other Areas

There may be debate over what formulas best capture a player's luck andskill, just as there is debate over whether a baseball player's prowessis best measured by home runs, slugging percentage, on-base percentage,or runs-batted in. Perhaps there will be other luck and skill scoresproposed which use slightly different formulas than those illustratedabove. Although it is believed that the primary luck and skill scores asdescribed in the earlier section are the best mode for practicing theinvention, other luck and skill scores derived from different formulasmay provide additional insight as well. These other formulas are termed“secondary poker luck and skill statistics” herein, and are consideredpart of the present invention. As an example, a different scheme forcalculating poker luck, and the rationale behind it, is detailed below.

A player's luck score L_(n) for the n^(th) hand is as follows:

$\begin{matrix}{L_{n} = {1 - \frac{p - 1}{h - 1}}} & (13)\end{matrix}$

where

-   p=placing in the hand (1^(st), 2^(nd), 3^(rd), . . . ) if all the    players had played their cards until showdown (i.e., no one folds),    and-   h=the total number of players dealt cards in the given hand.

If there is a tie for a placing, then a player's luck score L_(n) forthe n^(th) hand is determined as follows. Assume that q players are tiedfor p^(th) place. Then each tied player's luck score is the average ofthe luck score for p^(th) place, (p+1)^(th) place, . . . , and(p+s−1)^(th) place:

$\begin{matrix}{L_{n} = {{\frac{1}{q}{\sum\limits_{i = p}^{p + q - 1}\; 1}} - \frac{i - 1}{h - 1}}} & (14)\end{matrix}$

Simplifying via the well-known relation

${\sum\limits_{j = 1}^{m}\; j} = \frac{m( {m + 1} )}{2}$

we obtain

$\begin{matrix}{L_{n} = {{1 - \frac{p + \frac{q - 3}{2}}{h - 1}} = {\lbrack {1 - \frac{p - 1}{h - 1}} \rbrack - \frac{q - 1}{2( {h - 1} )}}}} & (15)\end{matrix}$

Equation (3) demonstrates that the luck score of a player tied forp^(th) place is the luck score the player would have received if he wereuntied for p^(th) place, reduced by the factor

$\frac{q - 1}{2( {h - 1} )}.$

The player's luck score l for the period of interest, for example allthe hands of a single tournament, is then the average of all theindividual luck scores for each hand:

$\begin{matrix}{l = {\frac{1}{k}{\sum\limits_{n = 1}^{k}\; L_{n}}}} & (16)\end{matrix}$

where

-   L_(n) is the luck score for the n^(th) hand of k total hands.

Let us examine a concrete example. Consider the case of Players A, B, C,D, and E who play a hand of poker. Without regard for the action thatactually transpires (who stays in the hand until showdown and whofolds), we consider the relative strength of each player's final hand ifthey all stayed in the hand until showdown. We determine, by knowing allthe players' hole cards and all the board cards, that A, B, C, D, and Ewould have placed first, second, third, fourth, and fifth respectively.Applying equation (1) we find the following:

Player Luck Score A 1 B 0.75 C 0.5 D 0.25 E 0

Simply put, the first place player will always have a luck score of 1,the last place player will always have a luck score of 0, and theremaining players will be distributed at equal intervals between 0 and1.

If A and B have the same strength hands, which beat the same strengthhands of C, D, and E, then A and B can be said to have tied for firstplace, with C, D, and E tying for third place. Equation (3) then yields:

Player Luck Score A 0.875 B 0.875 C 0.25 D 0.25 E 0.25

Simply put, the luck score of A and B is the average of the luck scoresfor untied first and second place, and the luck score of C, D, and E isthe average of the luck scores for untied third, fourth and fifth place.

Several points should be raised about the features of this secondaryluck score. First, it is clear that a player's average luck score, inthe absence of cheating, will tend towards 0.5 with a variance thatdecreases as then number of hands increases. Expressed as a percentage,the primary luck score becomes more intuitive for the average pokerplayer or fan: the average luck score is 50%, the luckiest possiblescore is 100%, and the unluckiest score is 0%. It is clear that afternumerous hands, a professional poker player's average lifetime luckscore will be minimally different from 50%. Therefore, if theprofessional has a better than average winning record, this can beattributed to skill, because the luck score has evened out. However, aprofessional player's luck score at a tournament's final table, whichwould involve far fewer hands, is of significance because it may well bedifferent from 50%. It would be of great interest to correlate theprofessional's luck score at the final table with his placing in thetournament.

Second, it is important to note that the formula considers the strengthof the hands as if no one folded, regardless of the fact that oftenpoker hands are not played to showdown. The reason for this is that aplayer cannot claim that his cards are unlucky if he folds before he canbe the recipient of his lucky cards, even if folding was the prudentstrategy. In fact, if folding is the prudent strategy, then it may havebeen made so by a skillful opponent who bet in order to induce the fold.Conversely, one might consider a player who is dealt pocket aces morethan his statistical fair share to be lucky—but not if the pocket acesare always beaten at showdown by another player who doesn't fold andmakes a better hand! It is therefore simpler to avoid the issue of whosecards were better earlier in the hand and consider only the relativestrength of each player's final hand had it been played to showdown.

Third, the formulation of the secondary luck score only takes intoaccount how each player's hand places relative to the others. It doesnot take into account, for example, how much stronger the first placehand is compared to the second place hand. Nor does the secondary luckscore consider the absolute strength of a player's hand. The reason forthis is that it is unclear whether having a much stronger hand isluckier or not. If a player's hand is much stronger than his opponent's,then the player can make larger bets with more confidence and thus winmore money. On the other hand, if the opponent possesses a relativelyweak hand, the opponent is more likely to fold, denying the player achance to make a big win. It is therefore simpler to avoid this issueand consider only the relative placing of each player's hand.

It is arguable that this secondary luck score is not the bestquantitative measure of a player's luck, precisely because it does nottake into account factors like whose hand was the strongest on earlierstreets, and how much stronger the hand was. Also, to compute thesecondary luck score, even if the hand doesn't go to showdown, knowledgeof the board cards that would have been dealt is required. Thisinformation is not usually displayed to the poker player or viewer. Theprimary luck and skill scores are better compared to the secondary luckscore in both these respects.

Venues in Which to Use the Luck and Skill Scores

Televised Events.

If the event is televised, the poker luck and skill scores could bedisplayed alongside each player's name at the end of each hand, or theycould be displayed at the same time the current standings and chipcounts are shown. The commentator could then analyze the action usingthis information. For example, the commentator might tell the viewersthat “Bob the amateur wins a big pot this hand using a large amount ofskill and a lesser amount of good luck.” The commentator might observethat “Phil the poker professional has gone broke in this cash game;despite a positive amount of skill, he lost because of a greater amountof bad luck.” The luck and skill scores would thus help answer in anobjective way how skillful and lucky a player had been during the game.

In order to calculate the luck and skill scores, you have to know thehole cards dealt to each player, as well as the community cards that aredealt. If the game is on the internet, these data can easily be obtainedand computed. For a televised event, particularly the World Series ofPoker Main Event, the number of entrants (over 5,000 in 2005 and 8,000in 2006) and the number of hands played probably preclude complete datacollection. One possible solution is to start collecting this data onlyafter all but twenty players have been eliminated, and present toviewers the overall scores for these final tables. Alternatively, theluck and skill scores for individual hands (and not the overall scores)can be easily captured from featured tables earlier in the tournament.

Live Casino Games.

During a live casino game, data collection can be automated by using apreviously patented device, the “Card Game Dispensing Shoe with Barrierand Scanner” (U.S. Pat. No. 6,582,301 B2). The dispensing shoe canrecord the cards that are dealt. A device to record the amounts put intothe pot by the players and to record when they fold would also benecessary. Taken together with the data from the shoe, the overall luckand skill scores could be generated, and presented to the playersinvolved in a live casino game once the game is over.

Internet Games.

If the game is on the internet, all the necessary data is easilycaptured. At the end of a tournament or cash game, each player'sstanding, overall luck score, and overall skill score could becalculated and listed. Each player could then assess his own performanceby seeing how much skill and luck (or lack thereof) contributed to hissuccess or failure. It would be very interesting to analyze the datafrom numerous games to see how lucky the winners typically are, or todetermine which poker variants empirically require more or less luck towin.

Video Poker Machines.

These machines, described by others, essentially convert a conventionalpoker table (using a human dealer, real playing cards, and chips) to acomputerized, electronic facsimile. Players using these machines makebets and view their cards via computer terminals around the table.Players place these bets and view their cards the same way that they dowhile playing on the internet. These machines allow players tocongregate around the table while playing each other via the computer,without the need for a human dealer. A “home game” version of thesemachines is also possible, where each player holds a compact, easilyportable, computerized tablet which is wirelessly linked to the tabletsheld by his opponents. Data collection to calculate the luck and skillscores via these machines is as easy as it would be on the internet. Thescores could then be displayed to the players at the end of a cash gameor at the conclusion of a tournament.

The invention comprises the methods, apparatus and systems whichimplement the formulas described above, including computer-implementedsystems. Where the invention is carried out by means of computerapparatus, the invention encompasses suitable executable computerinstructions, including routines, subroutines, programs, objects, datastructures and the like that perform certain functions or manipulate orimplement the data of interest.

The invention can be practiced with any suitable combination ofprocessing, input/output devices, display devices, and/orgeneral-purpose or special-purpose processors or logic circuitsprogrammed with the methods of the invention. Such devices can include,for example, personal computers, servers, client devices, personal dataassistants (PDAs), hand-held devices, laptops, programmable electronics,computer networks, such as, for example, a personal computer network, amainframe, and a suitable distributed computing environment thatincludes any of the foregoing.

While the invention has been described with reference to specificembodiments thereof, it should be understood that the invention iscapable of further modifications and that this disclosure is intended tocover any and all variations, uses, or adaptations of the inventionwhich follow the general principles of the invention. All suchalternatives, modifications and equivalents that may become obvious tothose of ordinary skill in the art upon reading the present disclosureare included within the spirit and scope of the invention.

1. A method of quantifying a player's luck and skill in a game of chanceand skill, said method comprises: calculating luck and skill scores forthe player by using a formula or a set of formulae based on probabilityof the player winning hands of the game, pot sizes of the hands, andamounts put in the pots by the player; and using the luck and skillscores to quantify the player's luck and skill.
 2. The method of claim1, wherein the game of chance and skill is a poker game.
 3. The methodof claim 2, wherein the formula or set of formulae comprises$L = {\sum\limits_{\underset{streets}{all}}\; T}$$S = {\sum\limits_{\underset{streets}{all}}\; U}$ wherein L is theluck score; S is the skill score; T is change in expected value V on agiven street as a result of a card (or cards) being dealt, or as aresult of forced bets (blinds and antes); and U is change in expectedvalue V on a given street as a result of players' unforced action, afterall action is complete on that street.
 4. The method of claim 2, whereinthe formula or set of formulae comprisesV _(i) =C _(i) P _(i) −B _(i) $\begin{matrix}{L = \frac{( {V_{1} - V_{0}} ) + ( {V_{3} - V_{2}} ) + ( {V_{5} - V_{4}} ) + ( {V_{7} - V_{6}} )}{N}} \\{S = \frac{( {V_{2} - V_{1}} ) + ( {V_{4} - V_{3}} ) + ( {V_{6} - V_{5}} ) + ( {V_{8} - V_{7}} )}{N}} \\{{L + S} = {\frac{V_{8} - V_{0}}{N} = \frac{V_{8}}{N}}}\end{matrix}$ wherein V₀=expected value before blinds and antes areposted and before hole cards dealt V₁=expected value before actionpreflop (after blinds and antes posted and after hole cards dealt)V₂=expected value after action preflop V₃=expected value before actionpostflop V₄=expected value after action postflop V₅=expected valuebefore action on the turn V₆=expected value after action on the turnV₇=expected value before action on the river V₈=expected value afteraction on the river (the amount actually won or lost in the hand)C=probability of the player winning the hand P=pot size B=amount put inthe pot by the player L=luck score for the hand S=skill score for thehand, and N=normalization factor.
 5. A method of enhancing the publicviewing of a game of chance and skill which comprises generating luckand skill scores for a player by the method of claim 1 and displayingsaid scores to the public during said game.
 6. A method of enhancing thepublic viewing of a poker game which comprises generating luck and skillscores for a player by the method of claim 3 and displaying said scoresto the public during said poker game.
 7. A method of enhancing thepublic viewing of a poker game which comprises generating luck and skillscores for a player by the method of claim 4 and displaying said scoresto the public during said poker game.
 8. A method of improving aplayer's skill level at poker which comprises generating luck and skillscores for the player by the method of claim
 3. 9. A method of improvinga player's skill level at poker which comprises generating luck andskill scores for the player by the method of claim
 4. 10. A method ofenhancing the experience of a game of chance and skill on the internetwhich comprises generating luck and skill scores for a player by themethod of claim 1 and displaying said scores to the public or to theplayers after said game.
 11. A method of enhancing the experience of apoker game on the internet which comprises generating luck and skillscores for a player by the method of claim 3 and displaying said scoresto the public or to the players after said poker game.
 12. A method ofenhancing the experience of a poker game on the internet which comprisesgenerating luck and skill scores for a player by the method of claim 4and displaying said scores to the public or to the players after saidpoker game.